In the case without capital-skill complementarity the Cobb-Douglas produc- tion function is being used instead. This case is helpful for reproducing the results of Judd (1985). The model in row M1 is a replication of Judd’s model with a representative capitalist and worker, perfectly competitive markets and a single labour income tax rate. Therefore, the model in that row can replicate the result of zero optimal capital income tax rate.
Note that the model in row M1 makes the assumption that both agents can save and work. But in order for the model to have a unique solution is being assumed that only the capitalist (which is assumed to be the skilled agent) has access to perfect capital market ( s = 0), whereas the worker (which is assumed to be the unskilled agent) needs to pay a cost of holding capital. However, in this case the optimal solution indicates that the worker will choose not to hold capital at all.
This result can be shown taking into account the two Euler equations for
capital structures (A33, A37) of the two agents at the steady state:22 1 =rst(1 r) + (1 st) 2 sKsts
and 1
=rst(1 r) + (1 st) 2 uKstu
since the left hand side of the above equations is identical, the right hand side of the two equations should be equal:
rst(1 r) + (1 st) 2 sKsts =rst(1 r) + (1 st) 2 uKstu taking into account the fact that s= 0:
rst(1 r) + (1 st) =rst(1 r) + (1 st) 2 uKu 2 uKstu = 0
and since, u > 0, the unskilled agents will not hold any capital at all no matter how large is the value of u:
Kstu = 0
Therefore, setting i = 0, for i = s; or i = u, will always result to the capitalist and worker economy where only the capitalist holds capital.
Taking the above into consideration, Judd’s (1985) theorem 4 is reinforced because even if only one of the two types of agents has access to perfect capital markets the optimal tax rate on capital income is zero. Speci…cally, when either of the two agents has no cost of holding capital the other agent will decide not to hold capital at all and the economy will be populated with capitalists that work and save and workers that only work.
When the production function doesn’t exhibit capital-skill complemen- tarities the zero optimal tax rate on capital income will remain valid even
22It makes no di¤erence if the Euler equations for capital equipment are being chosen instead.
if the government can tax di¤erently the labour income of capitalists and workers as the model in row M5. But in this case the government will choose to increase the progressivity of the labour income tax rate by taxing more the labour income of the agents that hold capital. As a result the after-tax skill premium is lower than the exogenous case which means that the government redistributes income to workers (the agents that do not hold capital) through labour income taxes.
As a result, with a single labour tax rate for both agents and no cost of capital holdings for the skilled agents (no capital market imperfection), the optimal policy suggests that capital income tax rate will be zero and the labour income tax rate will increase to balance the budget constraint.
Thus the level of government consumption will remain constant, as in the exogenous …scal policy case. But in the case where the government can impose two di¤erent wage tax rates, it will choose progressive labour income tax rates.
The di¤erence in the productivity between skilled and unskilled agents a¤ects only the progressivity of the labour income tax rate and as a conse- quence the after-tax skill premium (see the di¤erence between the models in rows M5 and M6).
The progressivity that a model without capital-skill complementarity re- ports is at the range of 0.11-0.14 with an exemption of 0.24 for the case where all elements are present (model in row M4). The progressivity reported from these models is lower than what it is observed in the data, apart from the case where it is equal to 0.24.23
When the model assumes that both types of agents have access to im- perfect capital markets ( s and u are positive but s < u) the optimal tax rate on capital income becomes positive. This positive result on capital income tax rate is not a¤ected from any of the other elements considered earlier (see models in rows M3, M4, M7 and M8).
Therefore, for the case without capital-skill complementarity the neces-
23The data from the UK government in 2012 report that the progressivity of income tax rate between the high income earners and the rest of the population is approximately equal to 30%.
sary condition of a non-zero optimal tax rate on capital income is that both agents have a positive cost of capital holdings. This result indicates that the main driving force of the non-zero optimal tax rate on capital income, under no capital-skill complementarity, is the imperfection in capital markets through the cost of holding capital.